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at the a=. :\stat524\
1:2]
,1:2]
{
%% (^) t(sigrnae$vectors) %% R12 (^) solve(msqrt(Rll)
Br --) %% R21 %% (R22) )
[1] 0.
correlations
. P2 =.
Therefore . PI
Test the hypothesis Ho: II2=O at the a= p- ~ n- tt--(n-1-(p+q+1)/2)log(prod( chicrit-qchisq(O.95,pq)
tt [1] 44. chicrit [1] 12. = 12.5916. Since the hypothesis
With degree of freedom pq=6, at a=
Ho:}:;12=O,and conclude that we correlations.
Test for the significance of the first k .. Ho :Pj 0,p2 = k. H I :P2 * 0 tt1--(n-1-(p+q+1)/2)log(prod(1-0. tt [1] 2. Chicrit1-qchisq(O.95, (p-1)*(q-1» Chicrit [1] 5.
The chi-squarecritical value = 5.9915. Since the observed test statistics 2.3458 is less than the
chi-square critical value, we don't reject the null hypothesis and we conclude that only first
canonical correlation is significant.
b) Using standardized variables, construct the canonical variates corresponding to the
"significant" canonical correlation(s).
1
Canonical variates
arl-solve(msqrt(Rll))%*%(-Areigen$vectors
arl
%*% (Breigen$vectors[,l])
[1,
[2,
br] br] [,1] [1,] 0. [2,] 0. [3,] 0.
Suppose Z(l) = [ zi1) , zi1) ], and Z(2) =! zi2) , zi2) , zj2) ] are standardized variables. Let Z = [ Z(l) , Z(2)]',
then 01 = a;z(l) = .7689 zi1) + .2721 zi1) , ~ = b;Z(2) = .0491 zi2) + :8975 zi2) + .1900 zj2) are the first
pair of canonical variates.
c)Using the results in Parts a and b, prepare a table showing the canonical variate coefficients and the samplecorrelations of the canonical variates with their component variables
rhoulzl-^ Rll%*%arl rhoulzl (^) ..,./ L,.I.J
rhov1z2-R22%*%br rhov1z
] 0. ] 0. --sol
[,1] 689274 720729 'e (msqrt (R22 )
b3-so1ve(msqrt(R22) ) %% a4-so1ve(msqrt(Rll))%% (- b4-so1ve(msqrt(R22)) %*% A-cbind(al, a2, a3, a4) B-cbind(bl, b2, b3, b4)
, and P: =. , Z~2) ] are standardized
.04732 zi2) -
are the first pair of are for The sample canonical Suppose Z(I)= [ Z:I) , Z~I)Z~I) variables. Let Z = [ Z<1), Z<2)]', then 61= a;Z(I)= .0430z:1)-1. .7806 Z~2)+ .2567 Z~2)+ .6919 Z~2)- canonical variates. The canonical presented in -" A U k .Columns of matrix B
A.
AI AI ~ a B b' 1 b' 2 [1,] 0.4732661 -0. [2,] -0.7805809 -0. [3,] 0.2567028 -0. [4,] 0.6919168 0. [5,] -0.1451489 -0. [6,] -0.0703867 0.6255409 - [7,] 0.3127276 0. [8,] 0.3364251 0.
~
According to canonical ~ while ~ represents annoyance variable while V2 represents a smoking I, smoking 2 and contentedness. rJ4 is primarily annoyance, sleepiness, alertness,
2 and smoking 3 variable 1 and smoking 4 .rJ3 is primarily alertness and while V4 represents
respective sets of
the canonical variates. .3067 , and
4
[1,] 0.04295049 -1.
[2,] -1.16220375 -0. [3,] 1.37533027 -0. [4,] -0.89086250 1.
Whether the first canonical
variables is reflected in "
1 Ls^2 1 2 2
=- r. (2)=-[(,71986) +".+(.53232) ]=-(2.96362)=.3705. The first sample
8 k=l Vl.'i 8 8 canonical variate O 1 of the desire to smoke set accounts for 30.7% of the set's total sample variance. The first sample canonical variate ~ and physical state set accounts for 37.1 % of the set's total' -low proportions of sample variances, " of their respective sets of variables.
rhou1z1-R11%%a rhov1z2-R22%%b rhou1z [ ,1] 1 -0. 2 -0. 3 -0. 4 -0. rhov1z [,1]
~:i ~ ~, .~j?,
